List 2-distance coloring of planar graphs

Abstract

The $$2$$2-distance coloring of a graph $$G$$G is to color the vertices of $$G$$G so that every two vertices at distance at most $$2$$2 from each other get different colors. Let $$\chi _{2}^{l}(G)$$?2l(G) be the list 2-distance chromatic number of $$G$$G. In this paper, we show that (1) a planar graph $$G$$G with $$\Delta (G)\ge 12$$Δ(G)?12 which contains no $$3,5$$3,5-cycles and intersecting 4-cycles has $$\chi _{2}^{l}(G)\le \Delta +6$$?2l(G)≤Δ+6; (2) a planar graph $$G$$G with $$\Delta (G)\le 5$$Δ(G)≤5 and $$g(G)\ge 5$$g(G)?5 has $$\chi _{2}^{l}(G)\le 13$$?2l(G)≤13.